• Friday: Lab 3 & A3 due

• Mon Oct 1: Exam I this room, 12 pm Please, no computers or

smartphones

• Mon Oct 1: No grad seminar

Next grad seminar: Wednesday, Oct 10

Type II error & Power

Today

Table 7.1 Generic recipe for decision making with statistics

1. State population, conditions for taking sample2. State the model or measure of pattern……………………………ST3. State null hypothesis about population……………………………H0 4. State alternative hypothesis……………………………………… HA5. State tolerance for Type I error…………………………………… α6. State frequency distribution that gives probability of outcomes when

the Null Hypothesis is true. Choices:a) Permutations: distributions of all possible outcomesb) Empirical distribution obtained by random sampling of all possible

outcomes when H0 is truec) Cumulative distribution function (cdf) that applies when H0 is true

State assumptions when using a cdf such as Normal, F, t or chisquare7. Calculate the statistic. This is the observed outcome8. Calculate p-value for observed outcome relative to distribution of

outcomes when H0 is true9. If p less than α then reject H0 in favour of HA

If greater than α then not reject H010.Report statistic, p-value, sample size

Declare decision

Table 7.2 Key for choosing a FD of a statistic

Statistic of the population is a meanIf data are normal or cluster around a central value

If sample size is large(n>30)……....…………Normal distribution

If sample size is small(n<30)……....…………t distributionIf data are Poisson………………………………..Poisson distributionIf data are Binomial………………………………Binomial distributionIf data do not cluster around central value, examine residualsIf residuals are normal or cluster around a central value

If residuals are normal or cluster around a central valueIf sample size is large(n>30)……....…………Normal

distributionIf sample size is small(n<30)……....…………t distribution

If residuals are not normal………………………Empirical distribution

Statistic of the population is a varianceIf data are normal or cluster around a central value……...Chi-squareIf data do not cluster around a central value

If sample size is large(n>30)……....… …Chi-square distribution

If sample size is small(n<30)……....…………Empirical

Table 7.2 Key for choosing a FD of a statistic - continued

Statistic of the population: ratio of 2 variances (ANOVA tables)If data are normal or cluster around a central value…………….F distIf data do not cluster around central value, calculate residualsIf residuals are normal or cluster around a central value……….F distIf residuals do not cluster around a central value

If sample size is large(n>30)……....………………F distribution

If sample size is small(n<30)……....………………..…Empirical

Statistic is none of the aboveSearch statistical literature for apropriate

distribution or confer with a statisticianIf not in literature or can not be found…....………………..…Empirical

Example: jackal bones - revisited

Example: jackal bones - revisited

1.

2.

3.

4.

5.

Example: jackal bones - revisited

6. Key

7.

2

21

2121

11psnn

XXt

22

21

21

1ss

n

XXt

2

11

21

222

2112

nn

snsns p

Example: jackal bones - revisited

8. Calculate p from t dist

Example: jackal bones - revisited

9.

10.

Example: jackal bones - revisited

Is your data normal?

Example: jackal bones - revisited

Is your data normal?

It really does not matter!

The assumption is that the residuals follow a normal distribution

Example: jackal bones - revisited

Are your residuals normal?

Residuals

Fre

quen

cy

-5 0 5

01

23

45

Example: roach survival

Data:

Survival (Ts) in days of the roach Blatella vaga when kept without food or water

Females n=10 mean(Ts)=8.5 days var(Ts)=3.6 days

Males n=10 mean(Ts)=4.8 days var(Ts)=0.9 days

Is the variation in survival time equal between male and female roaches?

Data from Sokal & Rohlf 1995, p 189

Example: roach survival

1.

2.

3.

4.

5.

Example: roach survival

6. Key

7.

8.

Example: roach survival

9.

10.

Parameters

Formal models (equations) consist of variable quantities and parameters

Parameters have a fixed value in a particular situation

Parameters are found in

functional expressions of causal relations

statistical or empirical functions

theoretical frequency distributions

Parameters are obtained from data by estimation

Parameters - examples

1. Functional relationship. Scallops density

Mscal=k1 if R=5 or 6

Mscal=k2 if R not equal to 5 or 6

Mscal = kg caught pr unit area of seafloor

R = sediment roughness from 1 (sand) to 100 (cobble)

k = mean scallop catch

Red for params, blue for variables

Parameters - examples

2. Statistical relationship. Morphoedaphic equation

Mfish= 1.38 MEI0.4661

Mfish= kg ha-1 yr-1 fish caught per year from lake

MEI = ppm m-1 dissolved organics/lake depth

0.4661

1.38 kg ha-1 ppm-0.4661 m0.4661

Red for params, blue for variables

Parameters - examples

3. Frequency distribution. Normal distribution

Red for params, blue for variables

2

2

1

2

1)(

X

eYfpdf

XYwhere

Y

X

μ = mean

σ = standard deviation

Parameter estimates

1. Scallops density

Mscal= μ1 if R=5 or 6

Mscal= μ 2 if R not equal to 5 or 6

Theoretical model to calculate μ1 and μ2?

Non-existent

estimate from data recorded in 28 tows

Mscal= μ1=mean(MR=5,6) n=13

Mscal= μ2=mean(MR<>5,6) n=15

Parameter estimates

2. Ryder’s morphoedaphic equation

pM = α MEIβ

ln(pM) = + populationY = a + MEI ln(MEI) sample

XY

YYmean

XXYYnXYCov

XVarXYCov

MEI

MEI

ˆˆ

)(ˆ

))(()1(),(

)(/(),(ˆ

0

1

Statistical Inference

Two categories:

1. Hypothesis testing

Make decisions about an unknown population parameter

2. Estimation

specific values of an unknown population parameter

Parameters

Estimation:

1. Analytic formula

e.g. slope, mean

2. Iterative methods

criterion: maximize the likelihood of the parameter

common ways to measure the likelihood:

sums of squared deviations of data from model

G-statistic (Poisson, binomial)

Parameters

Uncertainty:

Confidence limit:

2 values between which we have a specified level of confidence (e.g. 95%) that the population parameter lies